Ling ZHANG, Dongli MA, Muqing YANG,*, Shoqi WANG

a School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China b Xi’an Modern Control Technology Research Institute, Xi’an 710065, China

KEYWORDS Energy balance;Lateral-directional stability;Multi-constrained optimization;Shading effect;Solar aircraft;Winglets

Abstract Installing winglets can notably improve the aerodynamic performance of solar aircraft.This paper proposes a multi-constraints optimization method of winglets for solar aircraft, aiming to enhance the corresponding uninterrupted cruising capability.An optimization objective function is formed and is separately studied in aerodynamic and structural terms.Qualitative analysis shows that the winglet design parameters are restricted by four special constraints(geometry,aerodynamics,energy and stability)of solar aircraft.The optimization process is constructed on the basis of a multi-island genetic algorithm, and carried out for a 15 m wingspan solar aircraft. Although the designed winglet is not as good as the traditional winglet in terms of drag and structural weight,the designed winglet provides a better 24 h cruising capability.The sensitivity between the objective function and the design parameters is investigated, and the winglet effects vary with respect to the wing aspect ratio (AR=10, 15, 19.6). The effect of the constraints is analysed quantitatively, and some basic laws are obtained.Moreover,the feasible design region and the possible optimal design parameters of winglets for different wing configurations are explored.The calculation results show that when the aspect ratio exceeds a certain value, the winglets will not benefit the aircraft.

1. Introduction

Solar aircraft have a 24 h uninterrupted cruising ability, and feature a higher aspect ratio, lower flight speeds and lower structural weight than traditional aircraft. Improvements in aerodynamic characteristics and appropriate reductions in structural weight are the key factors for the uninterrupted cruising of solar aircraft.1

Induced drag accounts for approximately 30% of the total drag in conventional aircraft,2and the generation of induced drag is closely related to the tip vortices caused by threedimensional flow at the wingtip, which can be explained by the Prandtl lifting line theory.3Research also shows that wingtip vortices are weaker but still exist on the wing with a large aspect ratio.

One of the most effective ways to reduce tip vortices is adding winglets with the capability of reducing vortex strength around the wingtip region. The influence of winglets on various attributes of aircraft has gradually been recognized over time. In 1976, Whitcomb4first systematically studied the design theory of winglets. The experiments showed that winglets can provide twice the lift-drag ratio increment as wing extension at the same cost of structural weight. After Whitcomb’s work, researchers began to explore new methods to reduce induced drag through wingtip devices with different layouts.3,5Moreover, people were aware that winglets also had structural effects,especially in the wing root region.Kravchenco’s study6focused on the additional bending moment caused by winglets, claiming that the incorporated winglets increased the bending stress at the wing root. Through many scholars’ efforts in this field, a traditional design method of winglets has been formed, which commonly involves increasing the lift-drag ratio with limited wing root bending moment,because such aircraft generally follow Breguet’s formula of range and flight time, which shows that both performance indexes are proportional to the lift-drag ratio.

Due to the rapid development of fluid calculation and numerical simulation technology,the potential impact of winglets on aircraft has been thoroughly studied. The effects of winglets with variable cant angles on aircraft manoeuvrability were reported in the work of Bourdin et al.7In this research,the authors demonstrated that a pair of winglets with an adaptive cant angle can provide a coupled control mechanism.Zhang et al.8carried out a numerical simulation of a whole aircraft with five different winglets,focusing on the effects of different winglets on the lateral static stability of aircraft. In the meantime, the application range of winglets has been further broadened.Panagiotou et al.9presented a winglet optimization procedure for Unmanned Aerial Vehicles (UAVs). The authors of the study concluded that the improvement in the lift-drag ratio was mainly caused by the increase in lift, rather than the decrease in drag. Liu et al.10,11performed a series of experiments and found that winglets have the ability to improve the aerodynamic performance of the wings with high aspect ratios.

Solar aircraft have different design methods and configuration characteristics from conventional aircraft,12and the winglets used in successfully flying solar aircraft are highly diverse.‘‘Zephyr-S” which broke the record for continuous flight time of solar-powered aircraft possesses two-section downward winglets with a small cant angle, whereas ‘‘Sunseeker Duo”adopts two-section upward triangle winglets.The solar aircraft with a flying wing configuration named ‘‘Aquila” adopts upward winglets with a large cant angle. The wingtip region of Korean ‘‘EAV-3” cannot be regarded as a winglet in the strictest sense, but can still be considered as a winglet with a cant angle of 0°. In addition, some solar aircraft such as‘‘Helios” and ‘‘Odysseus” do not have winglets.

Winglet design of solar aircraft demands comprehensive consideration of many factors. Firstly, the design of solar aircraft generally follows the energy balance and mass balance function,thus there is a strong coupling between energy,mass and aircraft performance in the close-loop system.13As an additional device in the wingtip region, winglets can easily break the balance. On the one hand, winglets bring about changes in aerodynamic performance of wings. On the other hand, the load-bearing condition of a high aspect ratio wing is particularly sensitive to tip aerodynamic changes. Secondly,the shading effect of winglets influences the light receiving area of solar cells,leading to a change of the input energy.‘‘Zephyr-S” avoids this problem by adopting downward winglets.Although ‘‘Owl” and ‘‘Aquila” adopt upward winglets, the outer wing of ‘‘Owl” is not equipped with solar cells, and the large sweep angle of ‘‘Aquila” weakens the shading effect.Thirdly, Li et al.14drew a conclusion that solar aircraft differ from conventional aircraft in lateral-directional aerodynamic derivatives due to the special configuration of the former,which creates differences in the helical mode and Dutch roll mode. Hence winglets are expected to have a significant impact on the lateral-directional stability characteristics of solar aircraft. All these influences create differences in the design process of winglets for solar aircraft. However, few studies in the literature have attempted to quantify these impacts,and neither of these effects has been incorporated into the optimization process for designing winglets for solar aircraft.

Therefore,the purpose of this paper is to establish a winglet design method for solar aircraft with specific constraints, and research the general law of such a design.This study is divided into four stages. In the first stage, the optimization model,which consists of five design parameters, one objective function and four constraints, is proposed. The objective function is decomposed into aerodynamic and structural factors, and the construction method for the constraints is explained in detail. In the second stage, the winglet optimization process for solar aircraft is constructed on the basis of a multi-island genetic algorithm. In the structural calculation, an efficient weight estimation method is simplified according to the characteristics of solar aircraft. In the third stage, an example of an optimal winglet design for a fixed-size solar aircraft is presented.The difference between this method and the traditional method is analysed by comparing the iteration process diagrams and numerical results. In the fourth stage, a sensitivity analysis is conducted,and a quantitative analysis of the objective function and the constraints is carried out. Then a more in-depth exploration is performed along with the variation in the aspect ratio of the wing. Based on the above analysis, feasible regions of design parameters and possible optimal design points for different wing configurations are discussed.The benefits of winglets with fixed parameters on wings with different aspect ratios are also analysed in this stage.The last stage illustrates guiding significance of this work by conducting a preliminary selection of effective winglets based on the geometric parameters of solar aircraft.

2. Optimization model

2.1. Design parameters

Fig.1(a)shows that,the shape of the winglet is determined by 5 design parameters: cant angle (φw), length (lw), sweep angle(Λw), taper ratio (λw) and twist angle (εwr). Among them, lwis related to the wingspan b, and λwis the ratio of tip chord length Cwtto root chord length Cwr.Refs.10 and 15 indicated that the length of winglets should not exceed 10% of the half wingspan.The range of design parameters is shown in Table 1.Moreover, it is defined that when the winglets point upward,φwis positive.

Transitional modification is carried out in the connecting area between wing and winglet to improve the local air flow.By combining the design parameters with different values,winglets with different configurations(see Fig.1(b))commonly used in solar aircraft are obtained.

Fig. 1 Parametric model of winglets.

Table 1 Range of design parameters.

2.2. Optimization objective

The energy balance for solar aircraft is shown in Fig. 2. Solar cells arranged on the wing surface convert light energy into electricity during the daytime. Approximately half of the energy is used for daytime consumption, whereas the remaining part is stored in batteries for night cruising; thus, daytime cruising generally will not be interrupted by a lack of energy.The necessary condition to complete 24 h of uninterrupted flight is that the energy can meet the requirements of night cruising, which means that the longer the cruising time that the fixed capacity batteries can achieve, the easier it is to accomplish the mission goal. Thus, the objective function is directly related to the required power for cruise during nighttime.

The energy in a solar aircraft satisfies the following equation:

Fig. 2 Energy cycle of solar aircraft.

where Esolaris the total input energy of the energy system,Edayand Ebattare the parts of Esolarused for daytime flight and stored in battery respectively, Eprop·dayand Eprop·loadare the parts of Edayused for propulsion system and airborne equipment respectively, Eprop·nightand Eload·nightare the parts of Ebattused for propulsion system and airborne equipment respectively, and ηbattis the transfer efficiency of the energy in batteries.

The duration of uninterrupted cruising without illumination is expressed as follows:

where ηmotorand ηpropare the efficiency of the motor and propeller respectively,and Pnightis the required power for cruise at night.

The flight altitude of the plane remains fixed during the day and night,which means that only polymer batteries are used to store energy. Pnightis expressed by the motor thrust T and cruising speed V as follows:

The force balance relationship exists as follows:

where α is the angle of attack, m is the mass of the whole aircraft, g is gravitational acceleration, L is lift, and D is drag.

Lift and drag are expressed as follows:

where ρ is the air density,S is the wing reference area,CLis the lift coefficient, and CDis the drag coefficient.

Note that α during the cruising phase is close to 0, so approximate assumptions can be made that cosα=1 and sinα=0. Pnightis expressed as a combination of Eqs. (3), (4)and (5) as follows:

Therefore, Pnight∝(m3CD2/CL3)0.5. Here (m3CD2/CL3)0.5is defined as the cruising power factor Pf, which can be divided into the mass factor Mfand the aerodynamic factor Af; this division is convenient for assessing the independent impact of two aspects.

Different from the objective function of traditional winglet optimization design,Eq.(7)regards structural mass as an independent factor rather than as a constraint,16as one of the multi-objectives,17or as one part of the objective function by assigning it a weight coefficient derived from experience.18

2.3. Constraints

2.3.1. Geometric constraint

At present, many solar aircraft19,20have omitted landing gear to minimize weight. These aircraft take off by manual projection or car assistance and land by body touchdown.The crosswind encountered during landing will create a roll angle on the airframe, leading to contact between the wingtip and the ground during landing and subsequently affecting the structural safety of the aircraft.

Fig. 3 shows half of the face view of a solar aircraft with a conventional configuration. When the airframe lands at a 0°roll angle,the lowest point of the fuselage touches the ground,and this point is named p.The tip point of the mid-wing before and after elastic deformation is named s1and s1′, respectively,and the corresponding point of the winglet is named s2and s2′.The angles ΦL1and ΦL2are defined as shown in Fig. 3. The relationship between the roll angle Φlanding,ΦL1and ΦL2determines the location of the landing point pl.

The geometric constraint is expressed as follows:

where Φ0is the minimum admissible value of ΦL2, which is determined by the rolling range of the aircraft at landing.

2.3.2. Aerodynamic constraint

As an additional device of the wing, winglets are mainly used to improve the local flow field at the wingtip; however, the addition of the winglets should not make the parameters of the aircraft deviate too much from the initial design.21Constraints can be imposed by limiting the amount of lift coefficient change, which is defined as ΔCL.

where CL,wingletis the lift coefficient of the aircraft with winglets, and CL,originis the lift coefficient without winglets.

The aerodynamic constraint is expressed as follows:

where ΔCL0is the maximum admissible value of ΔCL,which is determined by the aerodynamic design requirements.

2.3.3. Energy constraint

When adopting an upward winglet, shadows fall on some of the solar cells during the period when light illuminates the aircraft from the side, thereby reducing the energy gained during the daytime.

The solar radiation model proposed in Refs. 22 and 23 has been widely used in solar aircraft design and flight strategy research12,24. This model can be utilized to calculate the altitude angle αe, azimuth angle αzand radiation intensity Pscof the sun at any time in the year.Regardless of the change in roll angle φ and pitch angle θ,which means φ=0°and θ=0°,the change period of yaw angle ψ is consistent with the flight trajectory period t0.Note that the initial flight direction of the aircraft is towards the east and the right yaw is defined as positive. As shown in Fig. 4, the azimuth angle of the sun in the body axis system is as follows:

The unit vector of incident light in the body axis system is expressed as follows:

Fig. 3 Landing inclination angle.

Fig. 4 Solar radiation coordinate system.

Fig. 5 Solar cell node partition.

In calculating the shading effect of solar aircraft, scholars usually obtained energy loss by measuring the shadow area on the wing.26,27However,when winglets are installed,the shadow area on the wing will change frequently over time, resulting in a large increase in computation. Therefore, this paper uses the reverse projection method (Fig. 6) to solve this problem.The projection of the winglet corner point W1on plane Ω formed by the other three corner points W2,W3,and W4along the incident direction is W1′, and the projection of the centre pointon plane Ω is. As shown in Fig. 6, the vectors composed of two adjacent corner points are expressed as c1,c2, c3and c4respectively. The vectors from the corner points to pointare expressed as p1, p2, p3and p4respectively.

Whether a cell can be illuminated by light is determined as follows:

The same method is used to determine whether solar cells are shielded by other elements on the wing,and only those that are not shielded by the wing but shielded by the winglet are considered for energy loss.

The power loss of solar cells with serial number(i,j)caused by the winglets at a particular moment t (t1

Fig. 6 Reverse projection method.

where Sijis the light area of the solar cell numbered (i, j).

Daytime energy loss due to the winglet shading effect is expressed as follows:

The energy constraint is expressed as follows:

where E0is the maximum admissible value of Eloss, which is determined by the energy balance function.

2.3.4. Stability constraint

The configuration characteristics of solar aircraft cause the helical mode to diverge easily, and the modal damping of the Dutch roll mode is large.14The influence of winglets on modal characteristics needs to be quantified.

The aerodynamic derivatives of solar aircraft are calculated by the Vortex Lattice Method (VLM) program, and the lateral-directional aerodynamic derivatives of a solar aircraft and a general aviation aircraft are compared in Table 2.

When the winglets are installed, the above aerodynamic derivatives change as follows:

(1) Clβ, Cnβand CYβ: When the sideslip angle β>0, the velocity component of the airflow along the y axis can be further decomposed into the velocity along the wingspan and the velocity vertical to the wing surface.When adopting downward winglets, the asymmetric change in the angle of attack on both sides produces a right roll moment, and the difference in the drag produces a left yaw moment, so both Clβand Cnβdecrease. The opposite effects occur when adopting upper winglets. However, both configurations of the winglet lead to an increase in CYβ. (see Fig. 7)

Table 2 Comparison of lateral-directional aerodynamic derivatives.14.

(2) Clpand Cnp: When the roll angular velocity p>0, for both upward or downward winglets, the angle of attack of the right winglet increases and that of the left winglet decreases, causing left row and right yaw moments, so both Clpand Cnpincrease.

(3) Clrand Cnr: When the yaw angular velocity r>0, the airflow velocity of the left winglet increases, whereas that of the right winglet decreases. The component of the lift difference in the z direction leads to a right roll moment; thus, Clrincreases. The component of the lift difference in the y direction generates a yaw moment.However, the direction of the yaw moment depends on the relative position of the centre of gravity and the aerodynamic focus of the winglet. In addition, the drag difference leads to a left yaw moment. Therefore, the increase or decrease in Cnris uncertain.

Ignoring the effect of inertia, the approximate estimation formula of the eigenvalue of the helical mode is as follows:28

where Lβ,Lr,and Lpare the derivatives of the rolling moment with respect to the sideslip angle,yaw angular velocity and roll angular velocity, respectively, and Nβ, Nr, and Npare the derivatives of the yaw moment with respect to the sideslip angle, yaw angular velocity and roll angular velocity,respectively.

The denominator of the formula is generally greater than 0,whereas molecules can be positive or negative due to different aerodynamic derivatives. Thus, the criterion of the helical mode convergence can be written as follows:

When winglets are installed, the increase in Clris not conducive to modal convergence, and the changes in the other three values of Clβ, Cnβand Cnrare uncertain, thus the modal characteristics may be improved or worsened.

The approximate expression of the damping ratios of the Dutch roll mode is expressed as follows:28

The installation of winglets creates uncertain changes in Cnβ, Cnr, CYβand CYr, causing variation in ξn,dr.

The stability constraint is expressed as follows:

where C0is the minimum admissible value of Chelicaland, ξ1and ξ2are the boundary values of ξn,dr.These values are determined by flight quality requirements.

2.4. Summary

In summary, the optimization model is described as follows:

3. Optimization process

3.1. Optimization method

The design of a winglet involves five parameters,the combination of which may cause optimization to fall into a local optimum. The multi-island genetic algorithm (MOGA-II)29uses the multi-search elite method with high efficiency and intelligence; this approach does not prematurely converge to the local optimum, and makes the fitness of the new generation larger than that of the parent generation. Therefore,MOGA-II is adopted for the design optimization of the winglet. The optimization process is shown in Fig. 8. The optimal Latin hypercube method was used to generate 15 individuals in first generation with the estimation of calculation amount and computational time.

3.2. Aerodynamic calculation method

The Navier-Stokes(N-S)equation is used as the main governing equation to conduct aerodynamic calculations.

Fig. 7 Lateral airflow and lift on winglets.

Fig. 8 Optimization process.

Solar aircraft usually have low Reynolds number characteristics due to their low cruising speed and altitude;30therefore,the turbulence model is selected as the transition model coupled with the Shear Stress Transport (SST) turbulence model proposed by Langtry and Menter.31-33This model combines the SST k-ω turbulence model with the γ-Reθtransition formula, possessing advantages of the two, and contains two additional transport equations:one is the equation of the local transition Reynolds number Reθ, which constitutes the criterion for predicting the initial position of transition, and the other is the equation for the intermittent factor γ,which is used to simulate the flow in the transition region.

The equation of the local transition Reynolds number Reθis expressed as follows:

where xjrepresents the j direction in Cartesian coordinate system, Ujis the component of local velocity in j direction, Pθtis the source term used to force the transported scalar R～eθtto match the local value of Reθt, σθtis the model constant which controls the diffusion coefficient, μ is the molecular viscosity,and μtis the eddy viscosity.

The equation of the intermittent factor γ is expressed as follows:

where Pγ1and Eγ1are the transition sources, Pγ2and Eγ2are the destruction and relaminarization sources, respectively,and σfis one of the constants for the intermittency equation.

Detailed parametric explanations of the equation are explained in the Refs. 31-33. A previous study34validated the turbulence model with a benchmark model and showed that this model was suitable for low Reynolds number aerodynamic calculations. The structural mesh of the whole aircraft for aerodynamic calculation is constructed automatically by the reply control file at each iteration in the optimization process.

3.3. Structural calculation method

The Elham Modified Weight Estimation Technique(EMWET)35is adopted to estimate the impact of the winglets on weight.This approach ensures the calculation accuracy and can greatly shorten the calculation time.

EMWET models the wing-box skin, stringers, spar caps and webs with equivalent panels (Fig. 9), and applies elementary structural sizing methods to compute the amount of material required for those panels to withstand the applied loads.Unlike traditional aircraft, composite sandwich beams with circular and rectangular sections (Fig. 10) replace wing box structures in solar aircraft.36-38Both configurations of beams can be regarded as equivalent forms of wing box structures in the EMWET method, and the weight of the beams can be calculated with equivalent panels.

For the sake of weight reduction,solar aircraft usually use a flexible wing skin instead of a rigid skin,and the front and rear edges of the wing commonly comprise foam and thin carbon fibre layers and lack stiffness and strength38,39. Hence most of the weight of the wing is concentrated on the main beam,which is used to centrally convey the bending moment and torsional moment. A reasonable simplification of the structure is made during calculation:

Fig. 9 Equivalent flat panels located at an effective distance.

Fig. 10 Sections of wing beam of a solar aircraft.

(1) The wing beams are the only component of the wing that has the capacity to sustain loads.40

(2) The 0°layer on the panels and the±45°layer surrounding the beam are used to withstand bending and torsion,respectively.35

(3) The thickness of the layer varies continuously(not stepby-step) along the spreading direction.

(4) The weight of the winglet which is an integral shell bearing structure is proportional to the surface area.

The minimum required thicknesses of the upper and lower panels are expressed as follows:

where M is the local bending moment; σmaxuand σmaxlare the maximum permissible stresses in the upper panels and lower panels, respectively; ηtis the effective distance coefficient of the equivalent flat panels, for which the solution method is described in the Ref. 35; tmaxis the maximum thickness of the wing beam; and Suand Slare the actual widths of the upper and lower panels, respectively. In addition, two key parameters for the equivalent calculation are shown in Fig. 10: y is the distance between each element of the panels,and y0is the distance between the chord line and the neutral axis.

The thickness of the surrounding layer is calculated using shear flow q and the maximum allowable shear stress τmaxas follows:

The weight of the wing beam is expressed by three thickness parameters as follows:

where Cbeamis the width of the equivalent flat panels; ρpand ρware the densities of the panels and webs, respectively; and hfsand hrsare the heights of the front and rear webs,respectively.

According to simplification (4) listed above, the mass of a winglet is expressed by the area density σwingletand the surface area Swingletof the winglet as follows:

Part of the structural mass of the aircraft is not affected by the winglet,and this mass is recorded as mfixed.The total mass of the aircraft is expressed as follows:

4. Case study

4.1. Study object

The study object is a 15 m wingspan solar UAV with a conventional configuration. This aircraft is designed by the authors’team and is capable of carrying small monitoring equipment for 7-day uninterrupted cruising. The main dimensions of the aircraft are shown in Fig. 11, and the main design indexes are shown in Table 3.

The aircraft possesses a wing with a high aspect ratio and a large anhedral angle, which are typical features of solar aircraft. Therefore, it is valid to take this aircraft as the study object, and the research methods as well as the basic laws can also be used for reference by other types of solar aircraft. According to the design indexes of the aircraft, the

Fig. 11 Studied solar aircraft.

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values of the constraints are ΦL2=10°, ΔCL0=2.5%,E0=33.9 W·h, Chelical=1, ξ1=0.2, and ξ2=0.22.

4.2. Optimization results

After approximately 150 iterations,the optimal objective value began to converge(see Fig.12(a)).Fig.12(b)shows that when using traditional methods for optimization, more than half of the design values in the first three generations were abandoned because the bending moment of the wing root exceeded 3%.Thus, the design parameters can only be optimized within a limited range in the following subsequent generations. However, the overall impact of the final design value on the night endurance of the aircraft is not necessarily the best.

Fig. 13 shows that the winglet obtained by the multiconstrained design method differs from that obtained by the traditional optimization method in three aspects: cant angle,length and sweep angle.The cant angle is the design parameter with the most obvious change between the two methods.

Table 4 shows that the performance of the winglet obtained by the multi-constrained design method is not as good as that obtained by the traditional design method in terms of drag and weight.However,the winglet with a smaller cant angle is obviously superior to the other scheme in terms of lift. On the whole, the value of Pnightfor the winglet obtained by the multi-constrained design method is 3.06% less than that for the aircraft without winglets. The reduction in Pnightis 0.9%greater than that offered by the traditional winglet, which shows that the new winglet design is more conducive to improving uninterrupted cruising capability.

Fig. 12 Sections of wing beam of a solar aircraft.

Fig. 13 Optimal results from different design methods.

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5. Analysis

5.1. Sensitivity analysis

The correlation coefficient represents the influence of the design parameters on the objective function. The correlation coefficient ranges from 0 to 1, wherein the impact is greater as the coefficient approaches 1. The correlation coefficients of the five design parameters to Af, Mfand Pfare calculated.Fig. 14 shows that the cant angle and length are strongly correlated with Afbecause they directly affect the flow of the wingtip region. Concurrently, these two parameters also have a greater influence on Mfthan the other three parameters,because length largely affects the lift of the wingtip, and the cant angle directly determines the length of the arm between the additional lift increment point and the wing beam section.When Afand Mfare multiplied to Pf, the cant angle becomes the only dominant factor because when the length increases,the aerodynamic characteristics tend to be advantageous,whereas the structural characteristics tend to be disadvantageous, causing length to become a non-decisive factor. Therefore, the influence of the cant angle on the objective function and constraints is emphatically analysed.

Fig. 14 Correlation coefficients between design parameters and objective function.

5.2. Objective function analysis

The impact of the cant angle on CDis shown in Fig.15.When the cant angle is 20°or-20°,the turbulent vortices are concentrated in the tip region. The vortex core has a large crosssectional area and a slow dissipation velocity, which means that the winglets have a small weakening effect on the tip vortex.When the cant angle is 45°or-45°,the turbulent vortices distribute more centrally around the tip region, and the corresponding area decreases. Although larger turbulent kinetic energy appears at the last cross section, the total energy loss caused by the vortices is smaller. When the cant angle is-70°, the turbulent vortices are distributed along the winglet and the vorticity intensity decreases obviously, which means that the winglet of this configuration is more conducive to drag reduction. When the cant angle is 70°, the boundary layer accumulates on the turning region between the outer wing and the winglet, and the drag is greater.

In the cases of this section,the cant angle of the winglet has greater influence on its own lift than that of the wing. The smaller the cant angle is, the greater the normal component of the winglet surface in the z-axis; thus, the winglet with a smaller cant angle causes a greater lift increase and wing bending moment.

The winglet designed in Section 4 is different from common traditional winglets. Firstly, an aircraft with a conventional configuration tends to use winglets with larger cant angles.Secondly,an aircraft with a conventional configuration usually adopts an upward winglet design, rather than a downward design. By changing the aspect ratio from 10 to 19.6 with the same wing area,the influence of the cant angle on the objective function is further analysed. In this process, the area of the winglets, the ratio of the winglet length to the wingspan and the twist angle of the winglets remain unchanged.

Fig. 15 Turbulent kinetic energy of winglets with different configurations.

In Figs.16(a)and(b),the values of CLand CDhave similar trends with respect to changes in the cant angle under 3 different Aspect Ratios(AR).When the cant angle is small,the configuration of the winglet is not conducive to drag reduction,but the effect of lift enhancement is better. A large cant angle is beneficial to the diffusion of tip vortices and has a good drag reduction effect,but the lift component is concentrated on the y axis, so the lifting capacity is limited. When the aspect ratio of the wing is large,the induced drag proportion is small,leading to a relatively slow change in the drag coefficient with respect to the cant angle, but the lift effect obviously changes with respect to the cant angle. With the decrease in the wing aspect ratio, the induced drag takes a larger proportion, but the reduction in the aspect ratio of the winglets reduces the lift increase effect. Fig. 16(c) shows that the mass increase in the wing with a high aspect ratio due to the winglet is obviously higher than that of the wing with a low aspect ratio because the arm of the force is different. In addition, due to the 12°anhedral angle of the outer wing,the arm of the upward winglet is longer and gains more weight than the downward winglet at the same absolute value of cant angle.

Fig. 16(d) shows that when AR=19.6, the winglet mainly contributes to Pfby increasing lift. The gain from the increase in lift when the cant angle is small exceeds that from the decrease in drag and the loss due to the increase in mass,which makes the power gain from winglets with smaller cant angles larger than that from winglets with larger cant angles. When AR drops to 15, the variety of influences with different cant angles on the lift,drag and mass are offset by each other,causing the opening of the Pf-curve to become larger. When AR decreases to 10, ignoring the special aerodynamic characteristics of the winglet when the cant angle is 70°,the opening direction of the Pfcurve changes upward, which means that the effect of the winglets on drag is more critical than the corresponding effects on lift and mass.

5.3. Constraints analysis

5.3.1. Stability constraint

A total of 250 different winglets are generated by the Latin hypercube method, and the modal characteristics of aircraft with different winglets are calculated.

The main effect graph reflects the average response of a design parameter to the target value when it is randomly combined with all other parameters at each level. The interaction effect graph reflects the main effect of one design parameter on the target value when another parameter is set at both high and low levels.

Fig. 17(a) shows that the cant angle is the main factor affecting the convergence criterion of the helical modes, and the three parameters of length, taper ratio and sweep angle have little impact on Chelicalwith the change in levels.The average response value of Chelicalincreases significantly with the tip pointing from downward to upward, which is consistent with the analysis in Section 2.3.4. When the winglets adopt a large downward cant angle, the average response of Chelicalis less than 1, indicating the helical mode divergence. Fig. 17(a) also shows that the average effect of the length on Chelicalwith the change in levels is distinct.This phenomenon can be explained by Fig. 17(b). When the winglets point down, the criterion value changes with a slope which is 2 times that when the winglets point up.

Fig. 16 Variation in CL, CD, Mf and Pf with respect to cant angle under different aspect ratios.

Fig. 17 Effect graph of convergence criterion of helical mode.

Fig. 18 Effect graph of Dutch roll modal parameters.

The influence of the wing design parameters on the Dutch roll mode characteristic frequencies and damping ratios is more limited than that on the helical mode (see Fig. 18).Length is the main factor affecting the characteristic frequency, whereas the cant angle, length and sweep angle are the main factors affecting the damping ratio. Therefore, the stability constraint of the Dutch roll mode is relatively easier to satisfy.

5.3.2. Energy constraint

Fig. 19 Power loss in different trajectories.

When the cant angle is set to 63.3°,the winglet obtained by the multi-constraints design method in Section 4 causes 0.5%total energy loss.The power loss caused by the shading effect during the daytime along different tracks is shown in Fig. 19. When the aircraft flies on the south-north trajectory, during the sunrise and sunset period,the projection area of the winglet on the wing surface is large, but the small solar altitude angle indicates that the radiation power is lower; thus, there are two peaks with small amplitude on the curve. When the aircraft flies on the east-west trajectory, the winglets are projected onto the wing surface only near noon, and the solar altitude angle is also high, so the peak power loss occurs. When flying along the circular trajectory (t0=0.5 h), approximately periodic variations in aeand αZBoccur in each cycle. Power loss reaches a peak value locally when αZB=0°, whereas the peak value reaches a maximum of 14.3 W at 8:00 or 16:00, which is synthetically influenced by aeand αZB. The energy loss will reach a larger value if the cant angle or sweep angle is further increased.

Fig. 20 Principal effect graph of ΔEloss.

Fig.20 shows that the variety of twist angle has little effect on energy loss. As design parameters increasing, the influence of the sweep angle on energy loss decreases first and then increases,whereas that of the other three parameters increases continuously. It should be noted that the cant angle is limited to the positive interval. The length and taper ratio affect the area of the winglet,the cant angle affects the projection length along the span of the winglet, and the sweep angle affects the projection angle and the real length of the winglet.

5.4. Feasible region and optimal design point

The reduction in Pfwith respect to the cant angle when AR=19.6 is shown in Fig. 22. When the cant angle is 0°,the value Pfis the best, but the constraints limit the range of the cant angle. The stability constraint and geometric constraint limit the design scope of the large downward cant angle,the energy constraint limits the design scope of the large upward cant angle, and the aerodynamic constraint limits the design range of the small cant angle for both downward and upward layouts. Finally, the feasible design region is limited to two curve segments as shown in Fig.21.In the feasible region,when the cant angle is-20°,the optimization rate of Pfreaches 3.06%, which is used as the design value.

Fig. 21 Design domain of optimized interval.

Fig. 23 Applicability of winglet with different aspect ratio.

Ignoring the special flow phenomena, when the cant angle is large, the general law of the feasible region and the optimal value of the winglet for solar UAV can be obtained.Fig.22(a)shows that,when the aspect ratio of the wing is large,the curve opens downward,and the design points are determined by the aerodynamic constraint, which may be points 1 and 2. As the anhedral angle of the wing increases, the possibility of point 1 being the optimal value increases (see Fig. 22(b)). Fig. 22(d)shows that when the aspect ratio of the wing is small,the curve opens upward, and the design points are determined by the geometric constraint, energy constraint or stability constraint,which may be points 3, 4 or 5. As the anhedral angle of the wing decreases, the geometric constraint moves to the right,and the possibility that point 5 is the optimal value increases(see Fig. 22(c)).

5.5. Applicability of the winglet

The design range for the aspect ratio of solar aircraft is not limited to 20. Although a winglet with a small cant angle is more suitable for a wing with a high aspect ratio,Fig.23 shows that when the wing aspect ratio continues to increase,the ability of the winglet to reduce Pfdecreases.The optimized winglet is placed on different wings whose chord length and the ratio of outer-wing length to mid-wing length, whereas the wing aspect ratio changes due to the change in wingspan. With the increase in AR, the increase in Afis smaller, the increase in Mfis larger,and the optimized value of Pfdecreases.When AR=29, the improvement rate in Afand the gain rate of structural weight cancel out. At this time, the change rate of Pfis 0; hence, the winglets do not bring any benefit to the aircraft.When AR is greater than 29,the winglet will have a negative impact on the aircraft, which means it is not suitable to use the winglet. It can be inferred that when the cant angle is smaller, the boundary design point will move to the left, and when the cant angle is larger, the boundary design point will move to the right.

Fig. 22 Optimum cant angle of wings with different configurations.

6. Conclusions

(1) A downward winglet with a small cant angle is designed for the studied solar aircraft. Although the new winglet design has a 0.67% higher drag coefficient and 0.64%higher structural weight than the traditional design,the former has a 1.5%higher lift coefficient,which leads to a 0.9% reduction in the cruising power factor.

(2) The cant angle and length are found to be the main design parameters affecting the aerodynamic and structural factors, and the main design parameter affecting the uninterrupted cruising capability of solar aircraft is the cant angle.

(3) When the aspect ratio of the wing is greater than 15,the design points are determined by the aerodynamic constraint,and the winglets with small cant angles are better for the studied solar aircraft.When the aspect ratio of the wing is less than 10,the design points are determined by the geometric constraint, energy constraint or stability constraint, and the winglets with large cant angles are better.

(4) Winglets have a greater influence on the helical mode than on the Dutch roll mode of solar aircraft, and the cant angle is the main factor affecting the helical mode.The upward winglet is beneficial to the convergence of the helical mode, whereas the downward winglet may cause the divergence of the helical mode.

(5) Different flight trajectories cause different shading effects. When the studied aircraft navigates in a circular trajectory, the upward winglet with a large cant angle will cause more than 0.5%energy loss.When the energy surplus is small,the energy constraint should be considered in the design process.

(6) For winglets with fixed design parameters, as the wingspan increases, the aerodynamic optimization effect decreases, and the structural deterioration effect increases. When the two effects are offset, the winglets will not bring any benefits to the solar aircraft, and it is no longer suitable to install the winglets.